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10(1) (1999), 29–45. [EvdP-99] S. Evdokimov, I. Ponomarenko. On highly closed cellular algebras and highly closed isomorphisms. Electron. J. Comb. 6(1) (1999), R18. [EvdP-00] S. Evdokimov, I. Ponomarenko. Separability number and Schurity number of coherent configurations. Electron. J. Comb. 7(1) (2000), R31. [EvdPT-00] S. Evdokimov, I. Ponomarenko, G. Tinhofer. Forestal algebras and algebraic forests (on a new class of weakly compact graphs. Discrete Math. 225(1-3) (2000), 149–172. A. H. E. Muzychuk.

Bannai, T. Ito. Algebraic Combinatorics I. Benjamin/Cummings, Menlo Park, 1984. [Big-93] N. Biggs. Algebraic Graph Theory. 2nd edition, Cambridge University Press, 1993. [Bir-67] G. Birkhoff. Lattice Theory. Colloquium Publications, American Mathematical Society, Vol. 25, Providence, Rhode Island, third edition, 1967. C. Bose. Strongly regular graphs, partial geometries, and partially balanced designs. Pacific J. Math. 13 (1963), 389–419. C. M. Mesner. On linear associative algebras corresponding to association schemes of partially balanced designs.

Span(v1 , . . , vn ) = Ul = V. Then the matrix representation of X with respect to this basis is of the shape: X1,1 X1,2 . . X1,l 0 X2,2 . . X2,l .. .. . . . 0 0 . . Xl,l where Xi,i is the matrix representation of X on Ui /Ui−1 . From this, it follows that l χV (X) = l tr(Xi,i ) = i=1 χUi /Ui−1 (X). i=1 The radical of A is defined to be the set of all elements of A that are in the kernel of every irreducible representation of A. Algebras with trivial radical are called semisimple.